The return time of extreme events is an important characterization
of extremes. Figure REF b) shows the return time
of the anomalies of the 90 days temperature, estimated from the 1000
years long control run (black). With direct sampling clearly we can
not estimate correctly events with return time larger than a few centuries.
Thanks to the rare event algorithm, we can extend the estimate of
the return time curve to much rarer events. The return time can be
computed from the output of the rare event algorithm as described
in details in [1]} and [2]}.
The red line in figure REF b) has been obtained
by computing return time functions from the experiments with the algorithm
(the case \(k^{*}=20\)  and \(k^{*}=40\)  repeated twice with different
sets of initial conditions to improve the statistics) and averaging
the results in the areas of overlap. Using several experiments with
different values of \(k^{*}\)  and different sets of initial conditions
helps to improve the statistics and to obtain a better estimate. The
total computational cost of the experiments is of about 1090 years,
basically the same of the control run. We can see that the return
time curve obtained with the large deviation algorithm overlaps with
the upper part of the curve given by the control run (confirming the
correctness of the procedure), but extends to much larger values of
the return time. With the algorithm we can compute return times up
to \(10^{6}-10^{7}\)  years with a total computational cost of the order
of \(10^{3}\)  years. There is thus a gain of more than three orders
of magnitude in the sampling efficiency.
